Multistep Bloch-line-mediated Walker breakdown in ferromagnetic strips

A well-known feature of magnetic field driven dynamics of domain walls in ferromagnets is the existence of a threshold driving force at which the internal magnetization of the domain wall starts to precess—a phenomenon known as the Walker breakdown—resulting in an abrupt drop of the domain-wall propagation velocity. Here, we report on micromagnetic simulations of magnetic field driven domain-wall dynamics in thin ferromagnetic strips with perpendicular magnetic anisotropy which demonstrate that in wide enough strips Walker breakdown is a multistep process: It consists of several distinct velocity drops separated by short linear parts of the velocity vs field curve. These features originate from the repeated nucleation, propagation, and annihilation of an increasing number of Bloch lines within the domain wall as the driving field magnitude is increased. This mechanism arises due to magnetostatic effects breaking the symmetry between the two ends of the domain wall.

Figure 1

Schematic representation of the simulated system. Two out-of-plane polarized domains are separated by a DW, which in equilibrium is a pure Bloch wall. As illustrated in the figure, upon application of an out-of-plane magnetic field $B_{\text{ext}}<B_{\text{W}}$, the magnetization of the moving DW finds a steady-state orientation corresponding to a partial Néel wall structure (arrows), producing magnetic charges on the DW surfaces. To minimize the resulting magnetostatic energy, the DW tries to orient itself with the DW magnetization, leading to DW tilting.

Figure 2

(a) $v_{\mathrm{DW}}$ as a function of $B_{\mathrm{ext}}$ considering a representative subset of different $L_y$'s. Note the “smooth” velocity drop for narrow strips that changes first to a single large drop ($L_y=600\mathrm{nm}$) and then develops two or even three distinct velocity drops separated by short linear parts of the $v_{\text{DW}}\left(B_{\text{ext}}\right)$ curve upon increasing $L_y$. (b) All the simulated $v_{\mathrm{DW}}\left(B_{\mathrm{ext}}\right)$ data visualized as a contour plot, highlighting the nonmonotonic dependence of $B_{\text{W}}$ on $L_y$.

Figure 3

(a) An example of a typical $v_{\text{DW}}\left(B_{\text{ext}}\right)$ curve of narrow strips (here, the $L_y=90\mathrm{nm}$ case is shown), exhibiting a smooth velocity drop for $B_{\text{ext}}>B_{\text{W}}$. Panels (b) and (c) display space-time maps of the internal DW magnetization (with the color wheel indicating the mapping from colors to magnetization) for two $B_{\text{ext}}$ values (5 and 6 mT, respectively) as indicated in (a) with the two symbols. These describe the time evolution of the internal in-plane magnetization of the domain wall for different $y$ coordinates along the domain wall. The in-plane DW magnetization exhibits coherent (spatially uniform) periodic switching events, with the frequency of the events increasing with $B_{\text{ext}}$.

Figure 4

A closer look at the DW dynamics corresponding to a multistep Walker breakdown for a strip of width $L_y=1500\mathrm{nm}$. The $v_{\text{DW}}\left(B_{\text{ext}}\right)$ curve shown in (a) exhibits three distinct velocity drops, with the corresponding VBL dynamics illustrated by means of space-time maps of the DW internal in-plane magnetization (with the color wheel in the middle showing the mapping from colors to magnetization direction) in (b)–(d). Panel (b) shows a single VBL first nucleating from the bottom edge (i.e., the leading end of the DW just before the onset of Walker breakdown), and then traveling back and forth along the DW. These dynamics are responsible for the first velocity drop seen in (a). Panel (c) displays the VBL dynamics corresponding to the second velocity drop, where after an initial transient a VBL is first nucleated from the bottom edge, and shortly afterwards a second VBL is nucleated from the top edge. Their annihilation is followed by an almost immediate formation of another pair of VBLs that propagate to the edges, after which the process repeats. Panel (d) shows the dynamics corresponding to the third velocity drop, involving the simultaneous presence of three VBLs within the DW. Movies illustrating the dynamics shown in (b)–(d) are included as Supplemental Material [18].